ment 383 (2007) 25–40www.elsevier.com/locate/scitotenv
Science of the Total Environ
Modeling of SO2 scrubbing in spray towers
Amitava Bandyopadhyay a,b,⁎, Manindra N. Biswas a
a Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721 302, Indiab West Bengal Pollution Control Board, Alipore Regional Office, Industry House, 2nd Floor, 10, Camac Street, Kolkata, 700 017, India
Received 18 August 2006; received in revised form 6 April 2007; accepted 19 April 2007Available online 13 June 2007
The removal of SO2 from flue gases has receivedconsiderable attention over the years because of itsvarious deleterious effects to all forms of lives. Amongthe various physico-chemical wet and dry methods, wetmethod is considered to be the simplest and the mosteconomical method for gas scrubbing with very high
⁎ Corresponding author. West Bengal Pollution Control Board,Alipore Regional Office, Industry House, 2nd Floor, 10, Camac Street,Kolkata, 700 017, India. Tel./fax: +91 33 22823358.
removal efficiency. Thus compliance with SO2 stan-dards will in many cases result in the installation of wetscrubbers. Tomany (1975) concluded that despite someof its inherent shortcomings wet scrubber couldeffectively combat the gaseous pollution. Wet scrubberswith column internals are replaced with spray towers(Kohl and Reisenfeld, 1985) due to its ability to treatlarge volume of gas as also it offers (b) least pressuredrop (compared to any other gas–liquid contactingdevices) and hence is cost effective (b) a very high turndown ratio, (c) higher service factor, (d) smaller onsiteplot space, (e) no scaling problem as a flue gasdesulfurization scrubber, and (f) simpler operationthan any other air emission control device.
26 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
Literature indicates that commercial spray towers invarious forms have been investigated for the absorptionof SO2 in water and alkaline solutions over the decadesbut the realistic treatment for characterizing the spraytower does not seem to be available. Availableliteratures were reviewed some of which are brieflydiscussed here. Hixson and Scott (1935) reported on thegas-side mass transfer coefficient as a function of gasflow rate, liquid flow rate and tower height for SO2–H2O spray absorption system using droplet diameter inthe range of 1500 to 2000 μm, but drop size was notconsidered for characterizing the absorption processstudied. Although Johnstone and Williams (1939)reported on the rates of absorption of SO2 by fallingalkaline (NaOH) droplets of diameter ranging between2670 and 5970 μm, the outlet concentration of SO2 andthe liquid flow rate were not specified. Mehta andSharma (1970a,b) reported on the effects of gas flowrate, liquid flow rate, and tower height on the gas sidemass transfer coefficient as well as on the interfacial areaof contact in a spray tower using SO2–dilute NaOHabsorption system. But the inlet and outlet concentra-tions of SO2 were not reported. Furthermore, none ofthese studies reported on the spray tower removalefficiency as a function of the pertinent variables underconsideration. On the other hand, although Epstein et al.(1975) reported on SO2 removal in spray tower, they didremain silent about the droplet size. Operation ofCombustion Equipment Associates Prototype DoubleAlkali Process was reported (Kohl and Reisenfeld,1985) for SO2 scrubbing comprising a venturi scrubberfollowed by an absorber designed to operate as either atray or a spray tower. In this process, the gas wascontacted with a clear solution containing NaOH orNa2CO3 or Na2SO3 as the first alkali followed by a limesolution as the second outside the scrubber circuit toeliminate problems associated with scaling within theoperating scrubber. This investigation reported on theremoval efficiency, inlet and outlet SO2 concentrationsand flow rate. But droplet size and droplet velocity in thespray section were not reported.
It is conceivable from the foregoing discussion that thedroplet size smaller than about 250 μm (e.g., 50 μm–150 μm range) was not considered in these early studies,possibly due to the (i) shortcomings often associated withthe commercial atomizers to generate such smallerdroplets and (ii) chances of their significant entrainment.To overcome the difficulty of generation of small dropletswith uniform spray pattern the design features of thecommercial atomizers need be modified. On the otherhand, the entrainment could be restricted by increasing thedroplet relative velocity (e.g., ≈20–30 m/s) in the gas
stream and in the process such droplets could shear off thegas boundary layer significantly and reduce its thicknessas much as possible to achieve highmass transfer rates forthe gas side controlling systems. Operation of a spraysection using such small droplets for achieving highremoval efficiency of SO2 formed one of the objectives ofour study reported earlier (Bandyopadhyay and Biswas,2006) using water and dilute NaOH solution. An energyefficient and cost effective two-phase critical flowatomizer developed by Biswas (1982) was used forgenerating small droplets at very high velocity withoutsacrificing spray uniformity and entrainment. Thereported removal efficiency was correlated with dropletSauter Mean Diameter (SMD), droplet mean velocity,other flow variables and physico-chemical properties ofthe system studied. Besides reporting typical drop sizedistribution, droplet SMDs reported were 41.6 μm,72.2 μm, 102.3 μm, 119.8 μm and 139.8 μm havingrespective dropletmean velocities of 26.38m/s, 25.18m/s,25.90 m/s, 27.36 m/s and 22.06 m/s measured at adistance of 1m from the atomizer discharge end. The semi-empirical correlation developed for alkaline (NaOH)scrubbing indicated that the removal efficiency wouldincrease with the decrease in droplet SMD and also withthe increase in the reagent (NaOH) concentration. Thus,reducing the droplet size with decreased NaOH concen-tration could increase the removal efficiency and in that,the consumption of the reagent would be reduced. Veryhigh removal efficiency (≈100%) was achieved inthe system in alkaline scrubbing using dilute NaOHsolutions in the concentration range varied between 0.2and 5 mol/m3.
Critical appraisal of the existing literature on theoverview of the reagents that are being commerciallyused for SO2 scrubbing indicates that the selection of asuitable absorbent or scrubbing liquid poses a verycomplex problem for removing SO2 from waste gasstream and is of significant importance for processdesign as well. Many of the problems, experienced atvarious facilities, are the result of the misapplication of ascrubbing process. For instance, a limestone slurryscrubber that works well on steady and weak SO2
emitting stream generated in a coal-fired boiler will notbe suitable for stronger and fluctuating SO2 emittingstream produced by the metallurgical processes. Com-mon commercial scrubbing processes utilized lime/limestone slurry, spray dry scrubbing with lime, oncethrough sodium alkali, once through seawater and dualalkali. A brief overview of the scrubbing liquids that arebeing utilized commercially is presented below:
Lime and limestone slurry scrubbing is suitable forrelatively low concentrations of SO2 and moderate
27A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
collection efficiencies. They are characterized by veryhigh liquid circulation rates, most often in-situ forcedoxidation of sulfite to sulfate, can often be designed forzero effluent discharge and can produce a marketablequality of gypsum. Spray dry scrubbing (using lime) isapplicable to hot gases with a significant amount ofevaporative capacity. SO2 absorption occurs as waterevaporates leaving behind a dry mixture of sulfate andsulfite solids. Because final collection occurs in a BagHouse Filter or in an Electrostatic Precipitator, there is nosegregation of process and CaSO3/CaSO4 solids. Oncethrough sodium process uses soda ash or caustic andproduce a solution of soluble sodium salts. They are quitesimple and effective. Seawater scrubbers are actually aform of this, making use of natural alkalinity in seawater(sodium bicarbonate). While there are several types ofdual alkali processes in operation, they all use a solublescrubbing agent (usually sodium or aluminum based) thatis regenerated by reacting with another alkali (usuallylime). These processes offer the advantages of solutionscrubbing, a solid by-product and ability to handle a veryhigh concentration of SO2 at the inlet.
The details of applicability of various reactiveprocesses of SO2 scrubbing are furnished in Table 1 interms of reagents to be used, range of inlet SO2
concentration and percentage of removal efficiency ofSO2. It can be seen from the table that 99%+ removalefficiency can only be achieved by sodium basedscrubbing processes. Perhaps due to this reason, it hasbeen pointed out by Brady (1987) that the neutralizationcapacity of NaOH for SO2 is extremely high. The exitconcentration of SO2 below 5 ppmv (Threshold LimitValue of SO2) can best be achieved with sodium alkalisonly. Furthermore, when concentration of SO2 absorbedbecame sufficiently large to make the economics of asimple throwaway process uneconomical, the wastesolutions can be regenerated in the dual alkali process byreacting it with lime e.g., Ca(OH)2 outside the scrubbercircuit. This approach permits the gas to be contactedwith a clear solution of highly soluble salts, therebyminimizing scaling, plugging and erosion problems in
Table 1Details of applicability of SO2 scrubbing processes in alkaline reagents (Koh
Process Reagents Inlet
Lime slurry CaO b100Limestone slurry CaCO3 1000–Spray drying — lime CaO, Ca(OH)2 b100Dual alkali: lime sodium NaOH or Na2CO3; and CaO or Ca(OH)2 1200–Dual alkali: Dowa CaCO3 and Al2(SO4)3 1000–Once through seawater NaHCO3 (CaO) Up ∼Once through sodium NaOH or Na2CO3 b100
the absorbent circuit. The use of a clear reactivesolution, instead of a slurry as in case of lime orlimestone scrubbing, also offers potential for almostcomplete SO2 absorption rate because the SO2 removalreaction is not limited by the rate of dissolution of solidparticles. In addition, the dual alkali process has theability to handle a very high initial concentration of SO2
amongst all other processes reported so far (Table 1).In the light of the above findings, dilute NaOH
solution has been selected as a suitable reagent for thepresent study (which constitutes a part of the dual alkaliprocess) in order to attain very high removal efficien-cies. On the other hand, water has been chosen in ourpreludial study to analyze the system behavior in waterscrubbing since this will help in understanding theextent of enhancement of removal efficiency of SO2
while studying NaOH scrubbing. Water scrubbing thusforms a primary part of the overall NaOH scrubbing.
In the present article an attempt has been made todevelop simple realistic models for predicting theperformance of a spray tower ascribing similar sprayhydrodynamics reported earlier (Bandyopadhyay andBiswas, 2006) with a view to attain definite insight intothe process of absorption of SO2 in water and in alkalinedroplets and to validate the models with the datareported earlier. The prime objective of the present studyis to develop models that can easily calculate theremoval efficiency of SO2 both in water and in alkali(dilute NaOH) scrubbing using the pertinent variables ofan operating spray tower specially for droplet diametersranging between 40 and 150 μm moving at very highvelocity (≈20–30 m/s).
2. Modeling of the spray tower
In a spray column the soluble components from the gasphase are transferred to the liquid phase by continuouscounter-current contact of the two phases as the swarm ofdrops flowing downward and the gas flowing upward.The actual flow situation is very complex and normallydefies mathematical interpretation. Attempts are being
l and Reisenfeld, 1985; Hay et al., 2004)
concentration of SO2 (ppmv) By products Efficiency (%)
–6500 Calcium based solids 90–954500 Calcium based solids ∼95–3000 Calcium based solids 90–95150,000 Calcium based solids 99+25,000 Calcium based solids 85–982000 Calcium based solids ∼98–10,000 Na2SO3; Na2SO4 99+
28 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
made for developing models on the performance of SO2
scrubbing (in water and alkali) in a spray tower followedby analysis of predicted values.
2.1. Theoretical model for water scrubbing underlaminar flow
In the proposed model following assumptions arebeing made (Bandyopadhyay, 1996):
1. The shapes of all the droplets are spherical.2. No reaction takes place in the gas phase.3. The effect of pH on the droplet surface is neglected.4. Steady state operation of the spray column.5. Insignificant heat effects due to the physical mass
transfer (true as the heat capacity of water is largecompared to the gas).
6. The gas phase concentration is uniformly distrib-uted at the inlet and outlet of the column and thesame concentration gradient exists inside thedownward flowing drops that take into accountthe functionality of concentration with height.
7. The mass transfer decreases during down flow ofthe drops and is a function of time it is exposed tothe gas phase. The efficiency of the drop wouldalso vary with time.
8. Small diameter to length ratio of the spray column.9. Insignificant wall effect.
10. The flow of gas and droplets is regarded as plugflow with negligible gas phase back mixing.
11. Droplet breakup, droplet coalescence and multi-droplet interactions in a cloud are neglected.
12. The limiting case of Re=0 is unattended since sucha situation is rarely been realized in an operatingspray tower with very high droplet velocity.
13. Kinetics of SO2 absorption in water: Absorptionof SO2 in water is considered as a physicalmechanism (Lynn et al., 1955) with the followingreactions occur in the liquid phase,
SO2ðgÞ þ H2O↔SO2ðaqÞ ðAÞ
SO2ðaqÞ↔Hþ1 þ HSO−13 ðBÞ
HSO−13 ↔Hþ1 þ SO2−
3 ðCÞ
In this article the mathematical derivation forabsorption follows from the treatment reported indetail by Beg et al. (1991) for stripping considering themass transfer from the gas phase in the droplet surfaceand into the interior of it by diffusion instead of masstransfer from the droplet to the air by diffusion as in
stripping. Accordingly the concentrations are adjusted inthe derivation. A material balance in a section of thetower shows that under the above assumptions, theoverall removal efficiency is the same as the efficiency ofa single droplet and the overall mass transfer of SO2 intothe drop can be given by the following expression [seeAppendix for detailed derivation]
m ¼ 2:36DdjDRe1=2Sc1=3Cl 1� DRe1=2Sc1=3 þ ðC V=ClÞD V/ðFo VÞ
DRe1=2Sc1=3 þ RTqdHMw
D V/ðFo VÞ
!
ð1Þ
where, /ðFo VÞ ¼ 5:32Pln¼1
exp � 4k2n2D VtD2
dj
� �[see Appendix]
Eq. (xxia).The removal efficiency of gas (SO2) for the jth
droplet size range without any chemical reaction, can bedefined as
DNj ¼amount of gas ðSO2Þentering into the drop
amount of gas ðSO2Þflowing through a circle whose diameter is that of the drop
ð2Þ
In a spray tower, the gas (SO2) removal efficiency isthe result of the collective absorbing capacities of all thedroplets present in the tower. Therefore, the amount ofSO2 entering into the droplet can be evaluated byintegrating the rate of mass transfer of SO2 into thedrops over the whole time period during which thedroplets remain in relative motion with respect to thegas. Hence, Eq. (2) can be written as
gNj ¼
Z t
0mdt
½ðQGClÞ=nj� where; nj ¼ QL
ðk=6ÞD3dj
" #ð3Þ
Distinct droplet diameter from the array of dropletsmay be used in Eq. (3) by measuring the droplet sizedistribution, in conjunction with the number of dropletsavailable from the distribution analysis. Else, theaverage droplet diameter may be used in conjunctionwith the number of droplets derived from consideringthe homogeneous droplet flow model as used by Beget al. (1991). While using the homogeneous droplet flowmodel instead of considering the specific drop sizedistribution, a single droplet diameter (average) ischosen and the overall result entails an error as wasconsidered by Beg et al. (1991). The error has beeneliminated in the present model by considering thedistinct droplet diameter as Ddj and is used in Eq. (3)that covers all the droplet diameters existing in the arrayof droplets introduced into the spray tower.
In order to determine the spray tower removalefficiency it is necessary to normalize Eq. (3) with
29A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
respect to time. Such normalization can be done byassuming a dimensionless time τ as
s ¼ t=ðL=vgÞ ð4Þ
and dt ¼ ðL=vgÞds ð4aÞfor which the boundary conditions are at t=0, τ=0 andat t=L /vg, τ=1.
The spray tower removal efficiency with this conceptof normalization for jth droplet size range can bederived using Eqs. (1), (3), (4) and (4a)
gNj ¼2:36DdjDRe1=2Sc
1=3ClL
vg½ðQGClÞ=nj�Z 1
01� DRe1=2Sc1=3 þ C V
ClD V/ðFoWÞ
DRe1=2Sc1=3 þ RTqdHMw
D V/ðFoWÞ
!ds
substituting nj¼QL
ðk=6ÞD3dj
; vg ¼ QG
ðk=4ÞD2T
and on simplificationwe get
gNj ¼ 3:54LD2
T
D2dj
!QL
Q2G
� �ðDRe1=2Sc1=3Þ
Z 1
01� DRe1=2Sc1=3 þ C V
ClD V/ðFoWÞ
DRe1=2Sc1=3 þ RTqdHMw
D V/ðFoWÞ
!ds
ð5Þ
where ϕ(Foʺ) is similar to that of Eq. (xxia) [seeAppendix] in terms of τ.
Eq. (5) actually describes the removal efficiencywhen the scrubbing liquid is recycled through the spraytower. Similarly, the removal efficiency for oncethrough scrubbing of the liquid can be obtained byputting C′=0 into Eq. (5)
gNj ¼ 3:54LD2
T
D2dj
!QL
Q2G
� �ðDRe1=2Sc1=3Þ
Z 1
01� DRe1=2Sc1=3
DRe1=2Sc1=3 þ RTqdHMw
D V/ðFoWÞ
!ds
ð6Þ
Eq. (6) may be directly used for average dropletdiameter (e.g., droplet SMD i.e., the volume to surfacemean droplet diameter which is generally used for thepurpose of gas–liquid absorption studies) for calculatingthe overall spray tower removal efficiency of SO2.While the overall spray tower removal efficiency of SO2
for all droplet size ranges available from droplet size
distribution may be calculated from the followingexpression
gNO ¼Xj¼1
gNjnj ð7Þ
2.2. Theoretical model for water scrubbing underturbulent condition
The analysis of removal efficiency under turbulentflow field is the main objective of this section (Bürkholz,1989). The gas molecules are molecularly more randomthan the liquid molecules and their movement is generallybeen characterized by statistical process under turbulencedue tomixing. On the other hand, droplets in any sprayingdevice are characterized as distinct particles with specificdiameter and when the gas is constantly mixing underturbulence then droplets will no longer have distincttrajectories. Thus the gas scrubbing becomes a statisticalprocess in such a situation. The probability of removal ofSO2 under this condition is randomand the efficiencymaybe derived as follows:
½change of gas phase concentration due to scrubbing �¼ ½gas concentration�d ½fraction of mass transferred into doplet phase in time dt�
and fraction of mass transferred into droplet phase
¼ mass transferred to all the dropletsamount of mass available for transfer to all the droplets
¼ md njðQGClÞ ¼
mðQGCl=njÞ
Or; dc ¼ �cdm
QGCl=nj
� �d dt ð8Þ
Eq. (8) upon integration and combining with Eq. (5)yields
Z Cout
Cin
dcc¼ �
Z t
0
mQGCl=nj
� �d dt ¼ �gNj
or; lnCout
Cin¼ �gNj; or;
Cout
Cin¼ expð�gNjÞ;
or; 1� Cout
Cin¼ 1� expð�gNjÞ;
ð9Þor, ηT,Nj=1−exp(−ηNj)
and similarly : gT;NO ¼ 1� expð�gNOÞ ð9aÞ
30 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
Eqs. (9) and (9a) are analogous to the famous DeutschEquation for interpreting the exponential particle collectionefficiencies under random turbulent motion and is purelydescribed by a statistical probability. It may so happenwhile using the equation under laminar flowmodel that theremoval efficiency exceeds the value of 1.0 (i.e., 100%) thatis meaningless because such a value of percentage removalof more than 100% is not meaningful. But the exponentialnature of the equation under turbulent flow in such asituation can yield meaningful values within 100%. This isdiscussed in details latter.
Eqs. (6), (7), (9) and (9a) can be applicable to anygas–liquid spray scrubbing process without chemicalreaction based on the assumptions and boundaryconditions specified by the physico-chemical hydrody-namics of the system. It is evident from Eq. (6) that theremoval efficiencies for water scrubbing of SO2 and thelike gases are independent on the concentration of gas ineither of the phases probably so because chemicalreactions are neglected. The efficiency of removalmainly depends on the spray hydrodynamics, physico-chemical properties of the system, operating variables(i.e., gas and liquid flow rates), dimensions (i.e., heightand cross-sectional area) of the spray tower. The impactsof the assumptions are discussed later.
2.3. Theoretical model for reactive scrubbing
In order to incorporate the effect of the reagent (e.g.,NaOH) on the scrubbing of SO2 by a falling droplet, it isnecessary to make the following additional assumptionsas made in case of water scrubbing:
1. At the surface nearer to the gas, there will be a zonethroughout which the reagent will be expended.
2. The presence of a uniform reaction front valid at anytime.
3. The pollutant gas (SO2) diffuses across the reagentfree zone from the surface of the drop to the reactionfront and the reagent on the other hand, diffuses fromthe interior of the drop toward the reaction front.
4. The concentrations of both the pollutant and thereagent are zero at the reaction front.
5. Kinetics of alkaline absorption of SO2: The followingtwo reactions should be considered for the absorptionof SO2 by aqueous NaOH solution, in addition to thereactions (A), (B) and (C) mentioned earlier,
SO2ðaqÞ þ OH−1↔HSO−13 ðDÞ
HSO−13 þ OH−1↔SO2−
3 þ H2O ðEÞ
Both reactions (D) and (E) may be regarded asinstantaneous reactions (Hikita et al., 1977) and affectthe alkaline scrubbing of SO2. The overall reaction is
SO2 þ 2NaOH ¼ Na2SO3 þ H2O ðFÞThe ratio of the rates of mass transfer of the pollutant
gas with and without chemical reaction occurring insidethe drop is proportional to the respective removalefficiencies of the pollutant gas in the spray tower. Thefinal form can be expressed by (Bandyopadhyay andBiswas, 2006)
mCj
mNj¼ gCj
gNj¼ 1þ i
M VDWMWD V
HMw
RTqd
CWCl
� �ð10Þ
Therefore, the overall spray tower removal efficiencyof SO2 for the entire array of the dropelts and withoutscrubbing liquid being recycled can be given as
gCO ¼Xj¼1
gCjnj ð11Þ
Under turbulent condition the removal efficiency canbe expressed similar to Eq. (9) as
gT;Cj ¼ 1� expð�gNjÞ ð12Þ
and similarly : gT;CO ¼ 1� expð�gCOÞ ð12aÞ
Eqs. (11), (12) and (12a) can be similarly applicableto any gas–liquid spray scrubbing process withchemical reaction based on the assumptions andboundary conditions specified by the physico-chemicalhydrodynamics of the system. It is evident from Eq. (11)that the removal efficiency of alkaline scrubbing of SO2
and the like gases are strongly dependent on the initialgas phase concentration and the initial reagent concen-tration in the liquid phase. It can be further noted that theremoval efficiency would decrease with an increase inthe initial gas phase concentration while it wouldincrease with an increase in the initial reagentconcentration. The removal of SO2 arrived at in alkalinescrubbing using Eq. (11) or (12) has been considered 1.0(i.e., 100% removal efficiency) while it exceeds 1.0,since such a situation (i.e., percentage removal morethan 100%) is not practically possible.
3. Prediction of removal efficiency of SO2
A model study has been attempted on the scrubbingof SO2 in H2O and NaOH using Eqs. (6) and (11)respectively on a hypothetical spray tower operating
31A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
counter-currently for predicting its behavior as a functionof pertinent parameters of the system. Eq. (9) has beenused while predicting the performance of the spray towerunder turbulent condition in water scrubbing. Distinctdroplet diameter has been used as the study dropletdiameter for the purpose of model calculation. Thelength and diameter of the tower assumed are 2.0 m and0.2 m respectively having length to diameter ratio of 10and is comparable to the reported data ranging between 6and 30 (Schmidt and Stichlmair, 1991).
Eq. (6) was numerically solved subject to the boundaryconditions. It may be noted that this numerical integrationcannot be extended to zero time and hence evaluation oftime intervals will always entail a small error dependingupon the initial time interval selected for the computa-tional work. This error, however, could be kept small bychoosing a sufficiently small length of interval of τ nearτ=0. The initial value of τ assumed was in the order of10−8 for achieving the numerical solution realistically.
To analyze the performance of the tower, values of v∞are needed. The value of the droplet terminal settlingvelocity, for free fall under the influence of gravity, is tobe used for v∞ (Crawford, 1980). In our study, the dropletradius of concern is in the range of 40 to 150 μm, movingat a very high mean velocity (≈20–30 m/s) as discussedearlier within a finite length of the tower (2 m). Underthese circumstances, the terminal settling velocities arenegligible compared to the mean droplet velocities.Therefore, the mean droplet velocities are used in ourcomputational work without entailing any error.
The rise in temperature on the droplet surface due toabsorption of SO2 (for 100–1400 ppm) is found to be ofthe order of 10−7–10−9 0C, using the equation proposedby Danckwerts (1970) and the heat of absorption of SO2
in a saturated solution is considered to be 6.7 kcal/mol(Roth, 1935). Such heat effects, however, appeared to besignificant in atmospheric absorption process whererelatively larger drops (≈1000 μm) were moving atterminal settling velocities (Reda and Carmichael,1982). Further, insignificant evaporation loss of smalldrops moving at very high velocity (≈20–30 m/s) in atower (of L=2 m) may be attributed to negligibleexposure time (≈0.1 s–0.067 s). Therefore, the heateffects due to mass transfer is considered insignificantand is neglected in the present work.
The droplet surface can represent a resistance to SO2
especially if the droplet pH is low (≈3). Ostensibly, theatmospheric droplets with such lower pH values limitthe absorption capacity for SO2. The present work, onthe contrary, attempts to develop theoretical modelsusing water and dilute alkali for scrubbing of SO2 in aspray tower wherein the pH of the scrubbing liquid
would be much higher (for instance, 7 to 11) and thisphenomenon is entirely different from the reportedatmospheric absorption processes so far (Brimblecombeand Spedding, 1972; Liss, 1971). This justifies inignoring the effect of pH in the present work.Furthermore, the concentration of SO2 would increaseat the droplet surface due to absorption of SO2 with timeand consequently the diffusion fluxes would reduce.This phenomenon has however, been considered in ourmodeling (see Appendix Eqs. (xv) and (xvi)).
Wall effect and gas phase back mixing are known tohave influence over the operation of a spray tower. Walleffect is assumed insignificant in our modeling becausethe separation of wall flow from the spray region flowdoes not have any practical significance (Mehta andSharma, 1970a,b; Schmidt and Stichlmair, 1991).Actual wall flow cannot be properly differentiatedfrom the spray region, because of the splashing of liquidfrom the wall. On the other hand, gas phase back mixingis reported to be negligibly small under plug flowcondition in pilot scale units and it increases with thediameter of the column as in industrial units (Mehta andSharma, 1970a,b). The gas phase back mixing is,therefore, neglected in our study.
The internal circulation has been analyzed for SO2
absorption by a single water droplet falling at terminalsettling velocity (Chen, 2001a) and it has beenelucidated that the impact of internal circulation iscomparable to mass diffusion for finer droplets. On theother hand, it has been shown that larger drops (Dd
≥500 μm) are deformed into oblate spheroids, whichwith increasing size develop a flattened base andsubsequently a concave depression (Le Clair, 1972).Therefore, the area over which the tangential stress,exerted by the flowing gaseous stream, acts on the dropbecomes rapidly smaller with increasing drop deforma-tion. Thus increased drop deformation results indecreased internal circulation. It has further beenillustrated by Chen (2001b) that the internal circulationcan be observed significantly for droplets havingdiameter larger than about 500 μm while investigatingthe SO2 absorption in falling rain drops at terminalsettling velocity. Furthermore, drops of diameter greaterthan about 500 μm have a tendency to oscillate andsubsequently it has clearly been demonstrated byPruppacher and Beard (1970) that such oscillationtend to disrupt organized internal circulation. Internalcirculation on the other hand, does not play significantrole in affecting the drop size. Therefore, dropdeformation and internal circulation play roles simulta-neously in gas-droplet absorption process especially forlarger drops (Dd≥500 μm) and would not affect the
Fig. 1. Effect of droplet diameter on the predicted percentage removalof SO2 for different droplet velocities.
32 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
mass transfer greatly as their effects are compensating innature to each other. As discussed earlier, our primeinterests in this study are to predict the percentageremoval of SO2 through simple model and validation ofthe predicted values with the available reported data formuch smaller droplets (ranging between 40 and 150 μm)moving at very high velocities (≈20–30 m/s). Underthese circumstances, the droplet deformation and dropletinternal circulation do not seem to have significantinfluence on the mass transfer and thus ignoring theireffects has been justified.
In Eq. (1), Fo′, the Frossling number, is the masstransfer Fourier number (Crawford, 1980) for the regioninside the drop. It represents the internal mass transferwithin the drop (i.e., it takes care of the liquid sideresistance to mass transfer) while the mass transfer on thesurface of the drop from the bulk of the gas phase isrepresented by the term “DRe1/2Sc1/3” (which takes careof the gas side resistance to mass transfer). Therefore, theintegrand in Eq. (6) combines both the effects in tandem.Furthermore, in the present work, the droplet sizedistribution is considered and mass transfer is determinedby integrating with respect to the dimensionless timefrom 0 (zero) to τ. Eq. (6), hence gives the combinedeffects of droplet hydrodynamics during its stay in thescrubber excepting droplet interactions, for instance,droplet breakup, droplet coalescence and multi-dropletinteractions in a cloud (Silverman and Sirignano, 1994).
The values of the physical parameters used in themodels are as follows: Diffusivities of SO2 in air, waterand NaOH are calculated to be 1.32×10−5 m2/s, 2.13×10−9 m2/s and 2.58×10−9 m2/s respectively (Chapmanand Cowling, 1970; Wilke and Chang, 1955; Vinogardand Mc Bain, 1941). While calculating the diffusivity inNaOH, it is assumed that the ratio of diffusivities requiredin Eqs. (11) and (12) is independent of temperature andviscosity. The other values used in the model calculationare H=4.85×106 N/m2, ν=1.56×10−5 m2/s, R=8314 J/kmol K, ρd=1000 kg/m3 and T=305 K.
3.1. Effect of droplet diameter and droplet velocity
Fig. 1 shows the effect of performance of the systemin water scrubbing with drop size at different dropletvelocities. The figure exhibits four distinct regimes asdiscussed below:
3.1.1. Regime — I: 1000 μm≥Dd≥300 μmIn this regime, the percentage removal is slowly
increasing with decreasing droplet diameter for a fixedhydraulic loading (i.e., QL=1.11×10
−5 m3/s — thestudy liquid flow rate). The percentage removal is
increased from about 45–46% to about 54–57% indecreasing the droplet diameter from 1000 μm to300 μm. Increase in droplet velocity has a subtle effecton the percentage removal in this regime. It may beattributed to the fact that the surface area increases atslower rate with the reduction in droplet size that may notbe sufficient enough to yield high mass transfer. Theincrease in mass transfer in reducing the droplet diameteris attributed due to the gradual reduction in the liquid sideresistance. This regime is termed here as droplet leanregime and is an ineffective gas absorption regime.
3.1.2. Regime — II: 300 μm≥Dd≥30 μmIn this regime, the increase in percentage removal
with the decrease in droplet diameter and with theincrease in droplet velocities becomes very rapid. Thepercentage removal is increased from about 54–57% toabout 62–82% in decreasing the droplet diameter from300 μm to 30 μm. It may be attributed to the fact thatvery fast increase in surface area with the reduction indroplet size might have reduced the liquid sideresistance [effect of Fo′] and also a large number ofhigh velocity droplets might have reduced the gas sideresistance [effect of “DRe1/2Sc1/3”], thereby drasticallyreducing the overall resistance in the gas–liquidinterface. It could be also due to the dramatic reductionof the characteristic time for diffusion of SO2 in theliquid phase [Ddj
2 /π ·D′] (Seinfeld, 1986) with thereduction in droplet diameter that resulted in very fastincrease in percentage removal. This regime is termedhere as the dense droplet regime.
3.1.3. Regime — III: Dd≤30 μm, vd≤10 m/sIn this regime, increase in surface area with a further
decrease in droplet diameter is not sufficient to compensatethe decrease in mass transfer resistance owing to increasing
Fig. 2. Effect of superficial gas velocity on the predicted percentageremoval of SO2 for different liquid flow rates.
33A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
rigidity of the droplets resulting in the percentage removalof SO2 decreases. The mass transfer cannot be increasedfurther on decrease in droplet size for vd ≤10 m/s. Thismeans that maximum reduction in liquid side resistance hasbeen achieved under this condition. The reduction in gasside resistance is also becoming limited at vd=10 m/swhich, however, can be further reduced in increasing vdbeyond 10m/s as discussed latter. The overall resistances tomass transfer shifts solely to the continuous gas phase forvery small droplets (Dd≤30 μm), which behave as solid orrigid sphere. It can be further seen from the figure that theonset of droplet rigidity at vd=1.0 m/s, 5.0 m/s and 10 m/sappear at droplet diameters of 25 μm, 14 μm and 7 μmrespectively. Clearly, this demonstrates that besides thedroplet diameter, the droplet velocity is a strong function ofthe droplet rigidity. This regime is termed as the rigiddroplet (sphere) regime. Studies on the rigid droplet(sphere) have been a subject of significant importance sincelong back to a legion of researchers (Le Clair et al., 1972;Clift et al., 1978; Dutta et al., 1988). But all of themcharacterized the rigid droplet (sphere) either in terms ofdroplet Reynolds number or in terms of dimensionlessradius ratio. We have, however, obtained here specificdroplet size within the hydro dynamical regime studied forthe existence of rigid droplet (sphere) that does not seem toexist in the literature.
3.1.4. Regime — IV: Dd≤30 μm, vdN10 m/sIn this regime, the percentage removal of SO2 is
increased to a maximum (≈95% for vd 30 m/s; as can beseen from the figure) on increasing the dropletvelocities. Eq. (6) is dominated by Fo′ for droplets tobehave as rigid droplet up to a vd of 10 m/s (ca.); while[DRe1/2Sc1/3] dominates Eq. (6) for overcoming thedroplet rigidity at velocities somewhat greater than10 m/s thereby increasing the mass transfer. Thisdramatic behavior of the rigid droplets might be due tothe reduction in available residual gas-phase resistanceas discussed in Regime III earlier, possibly by theshearing off of the gas boundary layer by the highvelocity rigid droplets. This regime is termed as thedroplet inertia-controlling regime.
3.2. Effect of superficial gas velocity and liquid flowrate
The effect of superficial gas velocity on thepercentage removal in water scrubbing for differentliquid flow rates at a constant droplet velocity is shownin Fig. 2. It may be seen from the figure that at lowerliquid flow rates, the percentage removal increases verysharply than at higher liquid flow rates with the decrease
in superficial gas velocity. It is also seen from the figurethat the performance of the system remains almostinsignificant at lower liquid flow rates and at superficialgas velocities greater than 5 m/s. But, the performance isbecoming functional of superficial gas velocity evenbeyond 5 m/s at higher liquid flow rates and the changein percentage removal of SO2 is very slow. It may beattributed to the fact that the liquid side resistance isreduced drastically at high superficial gas velocitystemming from the intrinsic behavior of gas-dropletabsorption process described by Eq. (6) discussedearlier. The increase in removal efficiency with theincrease in liquid flow rate (at a constant dropletdiameter) might be due to the increase in droplet loadinginto the scrubber per Eq. (6). On the other hand,decrease in removal efficiency with the increase in vg upto about 5 m/s could be attributed to the lowering of thegas phase residence time and to its inverse variation withthe square of the gas flow rate per Eq. (6). The effect ofvg and QL on the percentage removal of SO2 alsoindicates that the percentage removal would increasewith the QL/QG ratio. The theoretical analysis on theeffects of vg and QL/QG ratio on the percentage removalof SO2 elucidated in this study support the experimentalobservations made by Schmidt and Stichlmair (1991).
The dotted line in Fig. 2 shows the percentageremoval calculated using Eq. (9) under turbulentcondition. It can be seen that at low liquid flow rates,the values calculated by Eqs. (6) and (9) correspondvery closely. However, at high liquid flow rate, forexample, at 5.13×10−4 m3/s, expectedly a noticeablereduction in the percentage removal under turbulentcondition is observed up to a superficial gas velocity ofapproximately 5 m/s. Owing to random turbulentmotion of the gas molecules and the liquid droplets
Fig. 4. Effect of droplet diameter and concentration ratio on thepredicted percentage removal of SO2.
34 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
the probability of removal of SO2 yields somewhatlower values than that obtainable under laminar flowas discussed while Eq. (9) is derived. It can furtherbe noted from the figure that the removal of SO2
−5 m3/s at vg of about 2.5 m/s. Whileturbulent model under these flow regimes achievesremoval of SO2 to a maximum of 100% due to theexponential nature of Eq. (9). Therefore, the applica-tion of these equations [Eqs. (6) and (9)] to a specificgas–liquid system can ascertain the flow behavior thatcan be further established by experimentation.
3.3. Effect of tower height
The effect of tower height on the percentage removal ofSO2 in water scrubbing for different liquid flow rates(1.11×10−5 m3/s and 1.39×10−5 m3/s) is shown inFig. 3. It is seen from the figure that the percentageremoval improveswith the increase in tower height, whichcan be attributed to the increased gas–liquid contact time.As expected, it can also be seen from the figure that theincrease in the liquid flow rate increased the percentageremoval of SO2 as discussed earlier. The change inpercentage removal of SO2 is significant up to a towerheight of about 3 m under the range of variables studied.
3.4. Effect of concentration ratio
In alkaline scrubbing of SO2 using NaOH solution,the effect of concentration ratio (CR) on the percentageremoval of SO2 for different initial droplet velocities isshown in Fig. 4. Expectedly, it can be seen from thefigure that the curves move upward on increasing the
Fig. 3. Effect of spray tower height on the predicted percentageremoval of SO2 for different superficial gas velocities and for constantliquid flow rate.
CR from zero (reagent free situation in water scrubbing)to 1.0, i.e., the percentage removal of SO2 would yieldhigher values in alkaline scrubbing than in waterscrubbing owing to the reduction in liquid sideresistance in alkali scrubbing. For example, removalefficiency of about 55% can be achieved forDd=200 μm, vd=1 m/s in water scrubbing (CR=0)while the removal efficiency becomes almost 90% inalkaline scrubbing (CR=1.0) under similar operatingconditions in the model scrubber. It can also be seenfrom the figure that the percentage removal of SO2 ofabout 86.25% can be obtained in alkaline scrubbingwith CR=1.0 for Dd=400 μm, vd=20 m/s while thesame removal efficiency can be achieved in waterscrubbing by using Dd=30 μm and same dropletvelocity (vd=20 m/s). Clearly, this indicates that thereduction in droplet diameter can reduce the reagentconsumption. Furthermore, the figure also reveals that95% removal of SO2 can be achieved for Dd=100 μm,vd=1 m/s in alkaline scrubbing (CR=1.0) while theremoval efficiency would increased to nearly 100% byincreasing the vd up to 5 m/s under other conditionsremained unaltered. Clearly, it demonstrates that theincrease in droplet velocity can also reduce reagentconsumption significantly. The improvement of removalefficiency in alkaline scrubbing demonstrates that thedesired performance can be adjusted by manipulatingthe different operating variables.
4. Experimental verification of the theoreticalmodels
The validation of the models has been performedwith the experimental data available in the literature(Bandyopadhyay and Biswas, 2006). The pertinent
Fig. 6. Comparison of predicted and experimental values of removal ofSO2 in alkali (dilute NaOH) scrubbing.
35A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
variables of experimentation in water scrubbing wereQG=3.35×10
−3 to 5.54×10−3 Nm3/s; QL=5.56×10−6
to 3.11×10−5 m3/s; C∞=500 to 1300 ppmv; L=2.0 m;DT=0.1905 m and T=305±1 K. The droplet velocitiesand droplet diameter were discussed earlier. Reporteddistinct droplet size distribution and other data enabledusing Eq. (7) for the purpose of predicting the removalefficiency in water scrubbing under laminar condition.Predicted values in water scrubbing are compared withthe experimental values reported and are presented inFig. 5. It can be seen from the figure that the laminarmodel fits the experimental data excellently well up toQL=1.11×10
−5 m3/s. While turbulent model is inexcellent agreement at QL= 1.83×10−5 m3/s. It istherefore, conceivable that the model predicts theexperimental values exactly under the assumptionsmade during modeling. It also indicates that the effectof internal circulation does not seem to have noticeableinfluence in such a situation. Furthermore, within thisrange of liquid flow rate, the effect of droplet break up,droplet coalescence and multi-droplet interactions in acloud might be mutually nullifying with each other sothat droplet interactions did not have noticeable impacton the overall scrubbing, resulting in predicted valuesagreeing excellently well with the reported experimentaldata. However, for QL≥1.83×10−5 m3/s, the experi-mental values were over predicted by the model(turbulent) (≈10%). It is evident from the figure thatthe larger the droplet SMD is the larger was thedeviation. It could be possibly due to the fact that dropletinteractions that were ignored in our modeling wereimportant for larger droplets in hindering mass transfer.
Fig. 5. Comparison of predicted and experimental values of removal ofSO2 in water scrubbing.
It may be explained as follows. The effect of the dropletbreakup and droplet coalescence are opposite in nature;i.e., droplet breakup enhances the overall mass transferdue to increased surface area while droplet coalescencereduces it owing to reduction in the surface area for masstransfer. The multi-droplet interactions in a cloud wouldenhance the gas-droplet absorption than that for anisolated droplet in the size range of 40 to 100 μm(Silverman and Sirignano, 1994). Furthermore, it hasalso been pointed out that this interaction with finerdrops (≈40 μm) has a strong influence than with largerdrops (≈100 μm). From the foregoing critical analysisof our model derived results coupled with theseavailable information in the literature on the effect ofdroplet interaction on gas-droplet mass transfer it can,therefore, be qualitatively concluded that dropletcoalescence over dominates amongst other factors notconsidered for the development of the model. It can alsobe seen from the figure that the deviation was increasedwith the increase in gas flow rate. Besides dropletinteractions as discussed earlier, it might be due to thefact that the droplet surface reached the equilibriumconcentration of SO2 rapidly at higher gas flow rate andcould not get enough time to diffuse into the bulk of thedroplet phase that hindered further absorption of SO2
into the droplet. Experimental findings thus revealedthat there was no noticeable internal circulationotherwise the situation could have been different fromwhat was observed. In contrast, at lower gas flow rate,the absorbed SO2 might get sufficient time to diffuseinto the bulk of the droplet phase and distributeuniformly within the droplet and as a result the masstransfer was greater than what it could be at higher gasflow rate. Thus the experimental findings indicate thatinternal circulation did not have considerable role within
36 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
the range of spray hydrodynamics analyzed under thepresent context, which supports the observation madeby Chen (2001a).
The various experimental parameters reported (Ban-dyopadhyay and Biswas, 2006) in alkali scrubbing wereQL=1.83×10
−5 m3/s [satisfying the turbulent conditionin the present work]; distinct droplet size distribution withdroplet SMD=102.9 μm, velocity of which was men-tioned earlier; QG=5.54×10
−3 Nm3/s; Cʺ=2×10−4 to50×10−4 kmol/m3; C∞=1 to 1318 ppmv and T=305.6±1 K. Experimental values reported were compared withthe predicted values and were presented in Fig. 6. It can beseen from the figure that the predicted values agreedexcellently well with the experimental values [≈5%deviation].
5. Conclusions
Simple realistic models were developed in order todescribe the gaseous removal process of SO2 byscrubbing with and without chemical reaction in spraytowers. Effects of droplet size, droplet velocity,superficial gas velocity, liquid flow rate and towerheight on the performance of such a system weretheoretically predicted for distinct drop size. Somestriking features were observed while predicting theperformance as functions of droplet diameter andvelocity in a hypothetical spray tower using the modelsdeveloped. Four distinct regimes, viz. droplet lean,dense droplet, rigid droplet and droplet inertia control-ling regimes, were found important in spray scrubbingprocesses. Model calculation also elucidated the exis-tence of rigid droplet (sphere) for a distinct droplet sizeat a specific droplet velocity that does not seem to beavailable in the literature. Theoretical studies indicatedthat the rigidity of the droplet was a strong function ofthe droplet diameter and the droplet velocity. Theoret-ical considerations further revealed that the spray towercould perform best in the droplet inertia-controllingregime. The effect of turbulence owing to mixingstemming from the random movement of the gasmolecules and the droplets was discussed as a part ofmodeling and it was expressed through an exponentialequation analogous to the famous Deutsch Equation. Itwas pointed out that the turbulent flow model wouldyield meaningful values of removal efficiencies maxi-mum up to 100% whence laminar flow model yieldedvalues more than 100% removal efficiencies. The flowbehavior could be ascertained by applying the equationsunder conditions of laminar and turbulent flow fields toa specific gas–liquid system, which could be establishedby experimentation subsequently. The model developed
for water scrubbing indicated that the removal of SO2
was found independent on the concentration of gas ineither of the phases for once-through use of thescrubbing liquid; whereas, it was found to vary withthe concentration of gas in both the phases when thescrubbing liquid would be recycled. Apart from theconcentration of gas, the removal efficiency was foundto be a strong function of spray hydrodynamics, flowrates and dimensions of the spray tower. On the otherhand, the model for alkali scrubbing indicated that theremoval of SO2 was dependent on both gas phaseconcentration and reagent concentration apart from thephysico-chemical and hydrodynamic parameters of thesystem. It was further noted that the removal efficiencywould decrease with the increase in the initial gas phaseconcentration while it would increase with an increase inthe initial reagent concentration. When the model-calculated value of the removal efficiency in alkalinescrubbing would exceed 1.0 (i.e., 100% removalefficiency), it was suggested to consider the value as1.0 since such a situation (i.e., removal efficiency morethan 100%) was not meaningful.
Distinct droplet diameters from the array of dropletswere used from the droplet size distribution inconjunction with the actual number of droplets presentin that array for validating the models with availableexperimental data. Predicted data agreed reasonably wellwith the experimental data at lower liquid flow rates withrelatively smaller droplet while it over predicted evenwith the turbulent model at higher liquid flow rates withrelatively larger droplet. The deviation between thepredicted and the experimental values was reported to bepredominantly due to droplet coalescence amongst otherfactors not considered during modeling. Similar was thetrend of deviation at higher gas flow rates for largerdroplet SMD and such deviation was reported to be dueto the effect of absorbed SO2 on the droplet surface,besides the droplet interactions mentioned earlier.Comparison of model predicted values with the availableexperimental data revealed that the internal circulationdid not have significant influence on the percentageremoval for small droplets having diameter rangingbetween 40 to 140 μm. On the other hand, the predictedvalues in alkali scrubbing were in excellent agreementwith the experimental values reported in the literature.Finally, the models developed could also be applied toany gas–liquid spray absorption process subject to theassumptions and conditions necessary to describe thespecific physico-chemical hydrodynamics of the system.However, it was suggested that further refinement couldbe made for taking the droplet interactions into accountin order to accurately predict the removal efficiency.
37A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
NomenclatureA surface area of drop available for mass transfer,
m2/m3
Cin gas concentration at the tower inlet, ppmv
Cout gas concentration at the tower outlet, ppmv
C′ initial pollutant gas concentration within thedrop, kg/m3
Cʺ initial reagent concentration in the drop, kg/m3
Cs′ concentration of pollutant gas inside the drop atits surface, in equilibrium to the concentrationof pollutant gas in air, kg/m3
Ct concentration of pollutant gas or reagent withinthe drop at any time and at any radial direction,kg/m3
C′t concentration of pollutant gas within thedrop at any time and at any radial direction,kg/m3
Cv concentration of pollutant gas in air, inequilibrium with the surface concentration ofpollutant gas inside the drop, kg/m3
C∝ bulk gas concentration far away from theconcentration boundary layer, kg/m3
D diffusion coefficient of the pollutant gas in air,m2/s
D′ diffusion coefficient of the pollutant gas inwater, m2/s
Dʺ diffusion coefficient of the reagent in water,m2/s
Ddj characteristic diameter of the jth droplet sizerange, m
DT spray tower diameter, mFo′ Frossling number (D′t /Ddj
2 ), dimensionlessFoʺ Frossling number in terms of dimensionless
time τ, dimensionlessH Henry’s law constant of the pollutant gas, N/
m2
i mole ratio of gaseous component (SO2) andalkaline reagent (NaOH), dimensionless
k mass transfer coefficient, m2/sL spray tower height, mm mass transfer rate, kg/smcj instantaneous mass transfer rate with chemical
reaction for jth droplet size range, kg/smNj mass transfer rate without chemical reaction for
jth droplet size range, kg/sM′ molecular weight of the pollutant gas, kg/kmolMʺ molecular weight of the reagent, kg/kmolMw molecular weight of water, kg/kmoln integer, dimensionlessnj number of droplet for jth size range,
dimensionlessQG gas flow rate, m3/s
QL liquid flow rate, m3/sr radial direction of the spherical droplet, mR universal gas constant, J/kmol KR1 radial co-ordinate along circumference of drop,
mRe Droplet Reynolds number ( vdDd j / ν),
dimensionlessSc Schmidt number based on physical properties
of air (ν /D), dimensionlessSh Sherwood number (kDdj /D), dimensionlesst contact time, sT temperature, Ku velocity, m/sU∝ velocity, m/svg superficial gas velocity, m/svd droplet velocity, m/sv∝ velocity of air far away from velocity boundary
layer, m/sx ordinate along x-axis (parallel to drop trajecto-
ry), my ordinate along y-axis (perpendicular to drop
trajectory), m
Greek lettersηcj removal efficiency of SO2 with chemical re-
action for jth droplet size range, dimensionlessηCO overall spray tower removal efficiency of SO2
with chemical reaction, dimensionlessηNj removal efficiency of SO2 without chemical
reaction for jth droplet size range, dimensionlessηNO overall spray tower removal efficiency of SO2
without chemical reaction, dimensionlessηT,cj spray tower removal efficiency of SO2 with
chemical reaction for jth droplet size rangeunder turbulent condition, dimensionless
ηT,CO overall spray tower removal efficiency of SO2
with chemical reaction under turbulent condi-tion, dimensionless
ηT,Nj spray tower removal efficiency of SO2 withoutchemical reaction for jth droplet size rangeunder turbulent condition, dimensionless
ηT,NO overall spray tower removal efficiency of SO2
without chemical reaction under turbulentcondition, dimensionless
β ratio of concentration boundary layer thickness tovelocity boundary layer thickness, dimensionless
38 A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
A. Derivation of theoreticalmodel forwater scrubbingunder laminar flow
Let us assume a drop moving through a mixture of airand SO2. The velocity of the air relative to the drop is v∝at a substantial distance from the drop; the concentrationof SO2 is C∝ at that point. A concentration boundarylayer will be formed next to the surface of the drop,across which the concentration will change fromessentially C∝ to Cv at the surface of the drop and Ct
is the concentration of SO2 within the drop at any timeand at any radial direction. The mass transfer rate is atmaximum when the droplet is initially formed, i.e., theyare in the vicinity of the nozzle. It decreases duringdown flow and, therefore, is a function of time it isexposed to the gas phase, which suggests that theefficiency of the drop would also vary with time.Analysis of the boundary layer, considering a cubicpolynomial for the concentration profile, leads to thefollowing expression for the concentration field
ðCt � CvÞ ¼ ðCl � CvÞ 32ydc
� 12
ydc
� �3" #
ðiÞ
The boundary layer thickness for axisymmetricbodies can be given as (Kays, 1966),
d ¼ 5:02m1=2
R1U3l
Z x
0U 5
lR21dx
� �1=2ðiiÞ
with the following conditions assuming potential flow,
Ul ¼ 32vlsinh; R1 ¼ Ddj
2sinh; x ¼ Ddj
2h ðiiiÞ
The instantaneous mass flux can be expressed as
mA¼ 3D
2bdðCl � CvÞ½where dc ¼ bd� ðivÞ
The mass transfer across an area element of sphericaldroplet is given by :
dm ¼ 3kDD2dj
4bdsinh dhðCl � CvÞ ðvÞ
where, Ddj is the characteristic diameter of the jthdroplet size range.
Eq. (v) on integration over the entire hemisphere ofthe droplet (θ: 0→π / 2) and on rearrangement under theabove hydrodynamic condition, we obtain
m ¼ 1:783D2
djD
bðCl � CvÞ vl
mDdj
� �1=2
ðviÞ
A mass balance about the control volume locatedwithin the concentration boundary layer around thedrop, gives the mass transfer rate m as:
m ¼ Cl
Z dc
02k
Ddj
2udy
�Z dc
0C V 2k
Ddj
2udy ðviiÞ
Integration of the above equation and substitution ofm from Eq. (vi) yields,
b ¼ 1:088ðScÞ�1=3 ðviiiÞSubstituting the value of β from Eq. (viii) into (vi)
gives,
m ¼ 1:64D2djDðCl � CvÞSc1=3 vl
mDdj
� �1=2
ðixÞ
The mass transfer coefficient, k, is given as:
k ¼ m
kD2djðCl � CvÞ ðxÞ
Combining Eqs. (ix) and (x) gives
Sh ¼ 0:522Sc1=3ðvlDdj=mÞ1=2¼ 0:522 Sc1=3Re1=2 ðxiÞThe constant in Eq. (xi) takes into account only the
mass transfer from the front half of the drop. Anempirical correlation for the overall mass transfer to asphere is considered (Geankoplis, 1972) here for takingthe mass transfer into the rear half of the drop with wakeeffects into account in the present model as given below
Sh ¼ 2þ 0:522Sc1=3Re0:53 ðxiiÞA comparison between Eqs. (xi) and (xii), consider-
ing the difference in exponents on the Reynolds number,suggests that a coefficient of 0.75 would be moreappropriate for Eq. (xi) to represent the overall masstransfer from the entire surface of the drop. Therefore,Eq. (xi) may be rewritten as
Sh ¼ 0:75Sc1=3Re1=2 ðxiiiÞUsing the modified coefficient of Eq. (xiii), Eq. (ix)
becomes,
m ¼ 2:36D2djDðCl � CvÞSc1=3 vl
mDdj
� �1=2
¼ 2:36DdjDðCl � CvÞSc1=3Re1=2 ðxivÞIn Eq. (xiv), Cv is time dependent and it should
be expressed as a function of time for finding the
39A. Bandyopadhyay, M.N. Biswas / Science of the Total Environment 383 (2007) 25–40
efficiency. When the drop is first introduced into the gasflow, its content of SO2 is uniformly distributed. As thescrubbing process proceeds, SO2 is absorbed into thedrop, and its concentration becomes higher at thesurface than that at points inside the drop. This effecthinders the absorption rate of SO2.
The absorption rate of SO2 can also be obtained fromthe concentration distribution within the drop as afunction of radial position and the time, under theboundary conditions prevailing at the surface of thedrop. Hence the mass flux is given by
mA¼ D V
dCt Vdr
jr¼Ddj
2
� � ðxvÞ
Accordingly the boundary condition can be ex-pressed as:
dCt Vdr
Ddj
2; t
� �¼ 0:75
DD V
Sc1=3vlmDdj
� �1=2
Cl � HMw
RTqdCt V
Ddj
2; t
� �� � ðxviÞ
A simpler and sufficiently accurate solution can beobtained if the concentrations at the interface of the dropare assumed to reach equilibrium immediately. Thesolution is then slightly adjusted to allow for the masstransfer rate into the drop as given by Eq. (xiv). Thegeneral solution under unsteady state condition is thengiven by
Ct Vðr; tÞ ¼ Cs Vþ Ddj
kðCs V� Ct VÞ 1r
Xln¼1
ð�1Þnn
sin2knrDdj
exp � 4k2n2D VtD2
dj
! ðxviiÞ
In the above equation the factor (1 / r) in the 2nd termof the right hand side has been correctly derived as thisfactor is shown as (1 /2) in the model presented by Beget al. (1991) [see their Eq. (27)].
Henry's law can correlate the two concentrations as,
Cv ¼ HMw
RTqdCs V ðxviiiÞ
Eq. (xvii) on differentiation yields.
dCt Vdr
jr¼Ddj
2
� � ¼ 4Ddj
ðCs V� C VÞXln¼1
exp � 4k2n2D VtD2
dj
!
ðxixÞ
Substitution of Eq. (xv) into Eq. (xviii) yields,
m ¼ 12:566DdjD VðCs V� C VÞXln¼1
exp � 4k2n2D VtD2
dj
!
ðxxÞSolving Eqs. (xiv) and (xx) simultaneously we obtain
2:36DdjDðCl � CvÞSc1=3Re1=2
¼ 12:566DdjD VðCs V� C VÞXln¼1
exp � 4k2n2D VtD2
dj
!
or
DSc1=3Re1=2ðCm � ClÞ
¼ 5:32D V C V� RTqdHMw
Cv
� �Xln¼1
exp � 4k2n2D VtD2
dj
!
putting Cv from Eq. (xviii) we get
Cv DSc1=3Re1=2 þ RTqdHMw
D V5:32Xln¼1
exp � 4k2n2D VtD2
dj
! !
¼ Cl DSc1=3Re1=2 þ C VCl
D V5:32Xln¼1
exp � 4k2n2D VtD2
dj
! !
or;Cv
Cl¼ DRe1=2Sc1=3 þ C V
ClD V/ðFo VÞ
DRe1=2Sc1=3 þ RTqdHMw
D V/ðFo VÞ ðxxiÞ
where;/ðFo VÞ ¼ 5:32Xln¼1
exp � 4k2n2D VtD2
dj
!ðxxiaÞ
Substitution of the (Cv /C∞) from Eq. (xxi) intoEq. (i), leads to Eq. (1) for the rate of mass transfer ofSO2 into the drop under the assumptions considered. Eq.(1) derived in the present mathematical model shouldnot have any concentration term outside the integral ofthe right hand side, which however, is not the case of Eq.(33) as derived by Beg et al. (1991). This has beencorrectly derived in our modeling.
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